Cruz, a 2016 presidential contender, sent a letter to the president Thursday saying the U.N. Security Council shouldn't take up the agreement until after Congress has been able to review and vote on the deal.

"I ask that you provide written assurances that you will take all necessary steps to block any UN Security Council resolution approving the JCPOA until the statutory timeline for Congressional review has run its course," Cruz said in the letter. "Until you provide such assurances, I intend to block all nominees for the Department of State and hold any legislation that reauthorizes funds for the Department of State."

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Cruz's letter comes as the Obama administration has circulated a draft resolution to the U.N. Security Council, with a vote expected next week.

Cruz said the deal states that no later than 90 days after the Security Council backs the deal, the United States will be required to "prematurely 'issue waivers' in order to roll back the U.S. sanctions regime in preparation for a future implementation date."

The Texas Republican said that the maneuver "assumes congressional consent of this agreement."

"Nowhere does the Annex mention the United States Congress or recognize mandatory Congressional review under the Iran Nuclear Agreement Review Act of 2015," he added. "It seems your Administration intended all along to circumvent this domestic review by moving the agreement to the UN Security Council before the mandatory 60-day review period ends, thus adopting an agreement without Congressional consent."

Cruz, a frequent critic of the Obama administration, isn't the only lawmaker worried.

In a separate letter sent to Obama on Thursday, Sens. Bob Corker (R-Tenn.), the chairman of the Foreign Relations Committee, and Ben Cardin (D-Md.), the committee's ranking member, asked the administration to delay a vote at the United Nations until after Congress considers the bill.

"We are deeply concerned that your administration plans to enable the United Nations Security Council to vote on the agreement before the United States Congress can do the same," the top members of the Foreign Relations Committee wrote. "Doing so would be contrary to your statement that 'it’s important for the American people and Congress to get a full opportunity to review this deal.' "

The White House sought to downplay lawmakers' concern that the United Nations Security Council vote could undermine the congressional review.

“The Security Council resolution does not lessen the importance of Congress or its review of the Joint Comprehensive Plan of Action," spokesman Eric Schultz told reporters aboard Air Force One. “We will not begin implementation of the plan until after the congressional review period is over.”

Asked whether the U.S. is locked into the agreement by the U.N., Schultz said "there is nothing in the Security Council resolution that requires the United States to take action to implement the agreement.”

Cruz's pledge to block State Department officials or funding isn't a new threat for the administration.